Methods for realistic and efficient simulation of moving objects

Abstract

A method is proposed to simulate a moving object with a changing orientation such as bending or twisting in a real-time computer graphics application. A Projective Dynamics local-global solver is adapted with discretized Cosserat object position and orientation constraints and potentials for the local and global solving steps. Rotatable rigid or deformable bodies may be simulated accordingly, such as for instance rods, with potential weights accounting for the material parameters and geometric properties of the objects, such as the radius, the mass density, the length, and/or, for elastic thin objects, the Young's modulus, thus enabling realistic simulation for different values. Furthermore, the proposed methods converge after a small number of iterations, independently from the mesh resolution, enabling fast implementation in a diversity of computer graphics simulation applications.

1 . A computer graphics method to render, with a processor, a moving object, wherein said method comprises:
 discretizing the object into a plurality of elements;   identifying, for each of the plurality of elements, a current position and orientation, and a current linear velocity and a current angular velocity corresponding to an object motion to simulate;   estimating, for each of the plurality of elements, a predicted position and orientation as a function of the current position, orientation, linear velocity and angular velocity;   formulating, for each of the plurality of elements, object motion constraints C i  corresponding to the object motion to simulate;   for each of the plurality of elements, projecting, with a local solver, the predicted element position and orientation into auxiliary projection variables p i  on the object motion constraints C i ;   estimating, with a global linear system solver, a refined predicted position and a refined predicted orientation for each of the plurality of elements of the moving object as a function of the auxiliary projection variables p i  for each of the plurality of elements and of the material and/or geometrical properties of the moving object; and   rendering each of the plurality of elements of the moving object at the respective refined predicted position in the computer graphics simulation.   
     
     
         2 . The method of  claim 1 , wherein the local and global solver steps are iterated several times. 
     
     
         3 . The method of  claim 1 , wherein discretizing the object comprises modeling the object as a Cosserat object with one or more elements. 
     
     
         4 . The method of  claim 3 , wherein the Cosserat object elements are associated with discretized position variables x∈   3  and discretized Cosserat object orientation quaternion variables u∈   4 . 
     
     
         5 . The method of  claim 3 , wherein the object motion constraints C i  comprise a Cosserat object bend and twist constraint C BT  as a function of a twist strain defined for each object element. 
     
     
         6 . The method of  claim 5 , wherein the local solver projection on the Cosserat object bend and twist constraint C BT  is optimized when the relative curvature between any pair of adjacent element orientations is zero. 
     
     
         7 . The method of  claim 3 , wherein the object motion constraints C i  comprise a Cosserat object stretch and shear constraint C SE  as a function of a stretch strain defined for each object element. 
     
     
         8 . The method of  claim 7 , wherein the local solver projection on the Cosserat object stretch and shear constraint C SE  comprises a first local optimization step, with the local solver, on the position variables and a second local optimization step, with the local solver, on the orientation variables, the first and the second steps being independent from each other. 
     
     
         9 . The method of  claim 8 , wherein the solution to the first local optimization on the position variables is reached when the element's differential positions have a unit length and are aligned with the normal of the Cosserat object's cross section, so as to preserve the element's length as in its initial configuration. 
     
     
         10 . The method of  claim 8 , wherein the solution to the second local optimization on the orientation variables is reached when the rotational difference between the normal of the Cosserat object's cross section and the tangent of the element is minimal. 
     
     
         11 . The method of  claim 3 , wherein predicting, with a global solver, the motion of the Cosserat object comprises calculating a matrix of weighted Cosserat potentials, the potentials being calculated as a function of the auxiliary projection variables p i  for the plurality of elements, and the weights being calculated as a function of the material and/or the geometrical properties of the Cosserat object. 
     
     
         12 . The method of  claim 11 , wherein the Cosserat object is a Cosserat rod and the geometrical properties of the Cosserat rod are defined by at least one of the radius of the rod and the length of the rod. 
     
     
         13 . The method of  claim 11 , wherein the Cosserat object is a Cosserat rod and the material properties of the Cosserat rod are defined by at least a mass density of the rod material. 
     
     
         14 . The method of  claim 13 , wherein the rod is elastic and the material properties of the Cosserat rod are further defined by the Young's modulus of the rod. 
     
     
         15 . The method of  claim 3 , wherein the Cosserat object is a Cosserat shell. 
     
     
         16 . The method of  claim 3 , wherein the Cosserat object is a Cosserat volume. 
     
     
         17 . The method of  claim 3 , wherein the object is rigid and the Cosserat object is a Cosserat point. 
     
     
         18 . The method of  claim 1 , further comprising calculating, for each of the plurality of elements, a predicted linear velocity as a function of the current position and the refined predicted position for each element, and a predicted angular velocity as a function of the current orientation and the refined predicted orientation for each element; and using the predicted linear velocity, the predicted angular velocity and the predicted refined predicted position and orientation as the current estimates for each element of the moving object in a next rendering calculation.